Turbine rotor blade airfoil profile

ABSTRACT

The present application provides a turbine rotor blade including an airfoil shape and a trailing edge shape. The airfoil shape may have a nominal profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in Table I, while the trailing edge profile is defined by Table II. The Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances in a unit of distance by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in the unit of distance (e.g., inches). The X and Y values, when connected by smooth continuing arcs, define airfoil profile sections at each distance Z. The airfoil profile sections at Z distances may be joined smoothly with one another to form a complete airfoil shape.

FIELD

The present disclosure relates to an airfoil for a turbine rotor blade of a stage of a turbine and, more particularly, to a shape defining a profile for an airfoil of a turbine rotor blade.

BACKGROUND

Some jet aircraft and simple cycle or combined cycle power plant systems employ turbines, or so-called turbomachines, in their design and operation. Some of these turbines employ airfoils (e.g., stator vanes or nozzles and rotor blades), which during operation are exposed to fluid flows. These airfoils are configured to aerodynamically interact with the fluid flows and to generate energy from these fluid flows as part of power generation. For example, the airfoils may be used to create thrust, to convert kinetic energy to mechanical energy, and/or to convert thermal energy to mechanical energy. As a result of this interaction and conversion, the aerodynamic characteristics of these airfoils may result in losses that have an impact on system and turbine operation, performance, thrust, efficiency, and power.

BRIEF DESCRIPTION

Aspects and advantages of the turbine rotor blades in accordance with the present disclosure will be set forth in part in the following description, or may be obvious from the description, or may be learned through practice of the technology.

In accordance with one embodiment, the present application provides a turbine rotor blade including an airfoil shape. The airfoil shape may have a nominal profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in Table I. The Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances expressed in a unit of distance by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in the unit of distance. The X and Y values, when connected by smooth continuing arcs, define airfoil profile sections at each Z value. The airfoil profile sections at Z values may be joined smoothly with one another to form a complete airfoil shape.

In accordance with another embodiment, the present application provides a turbine rotor blade including an airfoil having a uncoated nominal suction-side profile substantially in accordance with suction-side Cartesian coordinate values of X, Y and Z set forth in Table I. The Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances expressed in a unit of distance by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in the unit of distance. The X and Y values, when connected by smooth continuing arcs, define suction-side profile sections at each Z value. The suction-side profile sections at the Z values may be joined smoothly with one another to form a complete (uncoated) suction-side nominal shape.

In accordance with another embodiment, the present application provides a turbine rotor blade that includes an airfoil having a uncoated nominal trailing edge profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in Table II. The Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances expressed in a unit of distance by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in the unit of distance. The X and Y values, when connected by smooth continuing arcs, define trailing edge profile sections at each Z value. The trailing edge profile sections at the Z values may be joined smoothly with one another to form a complete airfoil trailing edge shape.

These and other features, aspects and advantages of the present turbine rotor blades will become better understood with reference to the following description and appended claims. The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments of the technology and, together with the description, serve to explain the principles of the technology.

BRIEF DESCRIPTION OF THE DRAWINGS

A full and enabling disclosure of the present turbine rotor blades, including the best mode of making and using the present systems and methods, directed to one of ordinary skill in the art, is set forth in the specification, which makes reference to the appended figures, in which:

FIG. 1 is a schematic representation of an exemplary turbine having multiple stages with each stage including alternating rows of turbine rotor blades and stationary vanes or nozzles, according to at least one embodiment of the present disclosure; and

FIG. 2 is a perspective view of a turbine rotor blade, according to at least one embodiment of the present disclosure;

FIG. 3 provides a cross-sectional view of an airfoil portion of the turbine rotor blade of FIG. 2; and

FIG. 4 is an enlarged view of a trailing edge of the airfoil portion shown in FIG. 3.

DETAILED DESCRIPTION

Reference now will be made in detail to embodiments of the turbine rotor blades, one or more examples of which are illustrated in the drawings. Each example is provided by way of explanation, rather than limitation of, the technology. In fact, it will be apparent to those skilled in the art that various modifications and variations can be made in the present technology without departing from the scope or spirit of the claimed technology. For instance, features illustrated or described as part of one embodiment can be used with another embodiment to yield a still further embodiment. Thus, it is intended that the present disclosure covers such modifications and variations as come within the scope of the appended claims and their equivalents.

The detailed description uses numerical and letter designations to refer to features in the drawings. Like or similar designations in the drawings and description have been used to refer to like or similar parts of the invention. As used herein, the terms “first”, “second”, and “third” may be used interchangeably to distinguish one component from another and are not intended to signify location or importance of the individual components.

As used herein, the terms “upstream” (or “forward”) and “downstream” (or “aft”) refer to the relative direction with respect to fluid flow in a fluid pathway. For example, “upstream” refers to the direction from which the fluid flows, and “downstream” refers to the direction to which the fluid flows.

It is sometimes necessary to describe parts that are disposed at differing radial positions with regard to a center axis. The term “radially” refers to the relative direction (i.e., movement or position) that is substantially perpendicular to an axial centerline of a particular component; the term “axially” refers to the relative direction (i.e., movement or position) that is substantially parallel and/or coaxially aligned to an axial centerline of a particular component; and the term “circumferentially” refers to the relative direction that extends around the axial centerline of a particular component.

Unless stated otherwise, terms of approximation, such as “generally,” or “about” include values within ten percent greater or less than the stated value. When used in the context of an angle or direction, such terms include within ten degrees greater or less than the stated angle or direction. For example, “generally vertical” includes directions within ten degrees of vertical in any direction, e.g., clockwise or counter-clockwise.

Referring now to the drawings, particularly to FIG. 1, an exemplary turbine 10 of a gas turbine 12 includes a plurality of turbine stages arranged in serial flow order. Each stage of the turbine includes a row of stationary turbine nozzles or vanes (e.g., nozzles 100) disposed axially adjacent to a corresponding rotating row of turbine rotor blades 50. Four stages are illustrated in FIG. 1. The exact number of stages of the turbine 10 is a choice of engineering design and may be more or less than the four stages illustrated in FIG. 1. The four stages are merely exemplary of one turbine design and are not intended to limit the presently claimed turbine rotor blade in any manner.

The first stage comprises a plurality of turbine nozzles or vanes 100 and a plurality of turbine rotor blades 150. The turbine nozzles 100 are mounted to a casing 16 of the turbine 12 and are annularly arranged about an axis of a turbine rotor 14. The turbine rotor blades 150 are annularly arranged about and coupled to the turbine rotor 14. A second stage of the turbine 12 includes a plurality of turbine nozzles 200 annularly arranged about the axis of the turbine rotor 14 and a plurality of turbine rotor blades 250 annularly arranged about and coupled to the turbine rotor 14. The turbine nozzles 200 may be coupled to the casing 16 of the turbine 12.

As shown in FIG. 1, a third stage of the turbine 12 includes a plurality of turbine nozzles 300 annularly arranged about the axis of the turbine rotor 14 and a plurality of turbine rotor blades 350 annularly arranged about and coupled to the turbine rotor 14. The turbine nozzles 300 may be coupled to the casing 16 of the turbine 12. In particular embodiments, the turbine 12 includes a fourth stage, which includes a plurality of turbine nozzles 400 annularly arranged about the axis of the turbine rotor 14 and a plurality of turbine rotor blades 450 annularly arranged about and coupled to the turbine rotor 14. The fourth stage turbine nozzles 400 may be coupled to the casing 16 of the turbine 12.

It will be appreciated that the turbine nozzles 100, 200, 300 and 400 and turbine rotor blades 150, 250, 350 and 450 are disposed or at least partially disposed within a hot gas path 18 of the turbine 12. The various stages of the turbine 10 at least partially define the hot gas path 18 through which combustion gases, as indicated by arrows 20, flow during operation of the gas turbine 12.

FIG. 2 provides a perspective view of an exemplary turbine rotor blade 50 as may be incorporated in any stage of the turbine 12. For example, in one embodiment the turbine rotor blade 50 as presented in Table I below is representative of a turbine rotor blade 150 of the plurality of turbine rotor blades 150 of the first stage.

As shown in FIGS. 2 and 3 collectively, the turbine rotor blade 50 includes an airfoil 52 having a pressure-side wall 54 and an opposing suction-side wall 56. The pressure-side wall 54 and the suction-side wall 56 meet or intersect at a leading edge 58 and a trailing edge 60 of the airfoil 52. A chord line 62 extends between the leading edge 58 and the trailing edge 60 such that pressure and suction-side walls 54, 56 can be said to extend in chord or chordwise between the leading edge 58 and the trailing edge 60.

As shown in FIG. 2, the airfoil 52 includes a root or first end 64, which intersects with and extends radially outwardly from a base or platform 66 of the turbine rotor blade 50. The airfoil 52 terminates radially at a second end or radial tip 68 of the airfoil 52. The pressure-side and suction-side walls 54, 56 can be said to extend in span or in a span-wise direction 70 between the root 64 and/or the platform 66 and the radial tip 68 of the airfoil 52. In other words, each turbine rotor blade 50 includes an airfoil 52 having opposing pressure-side and suction-side walls 54, 56 that extend in chord or chordwise 62 between opposing leading and trailing edges 58, 60 and that extend in span or span-wise 70 between the root 64 and the radial tip 68 of the airfoil 52.

In particular configurations, the airfoil 52 may include a fillet 72 formed between the platform 66 and the airfoil 52 proximate to the root 64. The fillet 72 can include a weld or braze fillet, which can be formed via conventional MIG welding, TIG welding, brazing, etc., and can include a profile that can reduce fluid dynamic losses as a result of the presence of fillet 72. In particular embodiments, the platform 66, the airfoil 52 and the fillet 72 can be formed as a single component, such as by casting and/or machining and/or 3D printing and/or any other suitable technique now known or later developed and/or discovered. In particular configurations, the turbine rotor blade 50 includes a mounting portion 74 (such as a dovetail joint), which is formed to connect and/or to secure the turbine rotor blade 50 to the rotor shaft 14.

The airfoil 52 of the turbine rotor blade 50 has a nominal profile at any cross-section taken between the platform 66 or the root 64 and the radial tip 68, e.g., the cross section shown in FIG. 3. As shown in FIG. 4, the airfoil 52 may further define a trailing edge profile 61 proximate the trailing edge 60. For example, in one embodiment, Table II below is representative of the trailing edge profile 61 for a turbine rotor blade 150 of the plurality of turbine rotor blades 150. A “profile” is the range of variation between measured points on an airfoil surface and the ideal position listed in Table I or Table II. The actual profile on a manufactured turbine blade will be different from those in Table I or Table II (e.g., due to coatings), and the design is robust to this variation, meaning that mechanical and aerodynamic function are not impaired.

The X, Y, and Z values are given in Table I and Table II as percentage values of the airfoil height or span, as measured in any given unit of distance (e.g., inches). As one example only, the height or radial span of the airfoil 52 of turbine rotor blade 50 may be from about 1 inch to about 10 inches (about 2.5 centimeters to about 25 centimeters) or from about 5 inches to about 10 inches (about 12.5 centimeters to about 25 centimeters) or from about 7 inches to about 10 inches (about 17.5 centimeters to about 25 centimeters). However, it is to be understood that heights below or above this range may also be employed as desired in the specific application.

A gas turbine hot gas path requires airfoils that meet system requirements of aerodynamic and mechanical blade loading and efficiency. To define the airfoil shape of each turbine rotor blade airfoil, there is a unique set or loci of points in space that meet the stage requirements and that can be manufactured. This unique loci of points meet the requirements for stage efficiency and are arrived at by iteration between aerodynamic and mechanical loadings, enabling the turbine to run in an efficient, safe and smooth manner. These points are unique and specific to the system.

The loci that define the turbine rotor blade airfoil profile include a set of about 2,400 points with X, Y and Z dimensions relative to a reference origin coordinate system. The Cartesian coordinate system of X, Y and Z values given in Table I below defines the profile of the turbine rotor blade airfoil at various locations along its height. Similarly, the loci that define the turbine rotor blade trailing edge profile include a set of about 700 points with X, Y, and Z dimensions relative to a reference origin coordinate system. The Cartesian coordinate system of X, Y and Z values given in Table II below defines the profile of the trailing edge of the turbine rotor blade airfoil at various locations along its height.

Table I and Table II list data for a uncoated airfoil at cold or room temperature. The envelope/tolerance for the coordinates is about +/−5% in a direction normal to any airfoil surface location and/or about +/−5% of the chord 62 in a direction nominal to any airfoil surface location. In other words, the airfoil layout, as embodied by the disclosure, is robust to this range of variation without impairment of mechanical and aerodynamic functions.

A point data origin 76 is defined at or proximate to the root 64 at the leading edge 58. In one embodiment as presented in Table I and Table II below, the point data origin 76 is defined at or proximate to (above or below) a transition or intersection line 78 defined between the fillet 72 and the airfoil 52. The point data origin 76 corresponds to the non-dimensional Z value of Table I and/or Table II (presented below) at Z equals 0.

The Cartesian coordinate values for the X, Y and Z coordinates are set forth in non-dimensionalized units provided in Table I and/or Table II, although other units of dimensions may be used when the values are appropriately converted. The X, Y and Z values set forth in Table I and/or Table II are expressed in non-dimensional form (X, Y, and Z) from 0% to 100% of the height or span of the airfoil 52. As one example only, the Cartesian coordinate values of X, Y and Z may be convertible to dimensional distances by multiplying the X, Y and Z values by a height of the airfoil at the leading edge 58 and multiplying by a constant number (e.g., 100). As one example only, the Cartesian coordinate values of X, Y and Z may be convertible to dimensional distances by multiplying the X, Y and Z values by a height of the airfoil at the trailing edge 60 and multiplying by a constant number (e.g., 100).

To convert the Z value to a Z coordinate value, e.g., in inches, the non-dimensional Z value given in Table I and/or Table II is multiplied by the Z height of the airfoil in inches. As described above, the Cartesian coordinate system has orthogonally-related X, Y and Z axes, and the X axis lies generally parallel to a centerline of the rotor shaft 14, i.e., the rotary axis, and a positive X coordinate value is axial toward an aft, i.e., exhaust end of the turbine 10. The positive Y coordinate value extends tangentially in the direction of rotation of the rotor, and the positive Z coordinate value is radially outwardly from the base 66 toward the radial tip 68. All the values in Table I and/or Table II are given at room temperature and do not include the fillet 72 or coatings (not shown).

By defining X and Y coordinate values at selected locations in a Z direction normal to the X, Y plane, the profile section or airfoil shape of the airfoil 52 of the turbine rotor blade 50 at each Z distance along the length of the airfoil 52 can be ascertained. By connecting the X and Y values with smooth continuing arcs, each profile section at each distance Z is fixed. The airfoil profiles of the various surface locations between the distances Z are determined by smoothly connecting the adjacent profile sections to one another to form the airfoil profile.

The Table I values are generated and shown to four decimal places for determining the profile of the airfoil 52. Similarly, the Table II values are generated and shown to four decimal places for determining the trailing edge profile 61 of airfoil 52. As the turbine rotor blade 50 heats up during operation of the gas turbine 12, surface stress and temperature will cause a change in the X, Y and Z values. Accordingly, the values for the profile given in Table I and/or Table II define the “nominal” airfoil profile, that is, the profile of an uncoated airfoil at ambient, non-operating or non-hot conditions (e.g., room temperature).

There are typical manufacturing tolerances as well as coatings which must be accounted for in the actual profile of the airfoil 52. Each cross-section is joined smoothly with the other cross-sections to form the complete airfoil shape. It will therefore be appreciated that +/− typical manufacturing tolerances, i.e., +/− values, including any coating thicknesses, are additive to the X and Y values given in Table I and/or Table II below. Accordingly, a distance of +/−5% in a direction normal to any surface location along the airfoil profile defines an airfoil profile envelope for this particular turbine rotor blade 50 airfoil design and turbine 12, i.e., a range of variation between measured points on the actual airfoil surface at nominal cold or room temperature and the ideal position of those points as given in Table I and Table II below at the same temperature. The data provided in Table I and/or Table II is scalable (i.e., by uniform geometric scaling), and the geometry pertains to all aerodynamic scales, at, above and/or below 3000 RPM. The design of the airfoil 52 for turbine rotor blade 50 is robust to this range of variation without impairment of mechanical and aerodynamic functions.

TABLE I SUCTION-SIDE PRESSURE-SIDE X (%) Y (%) Z (%) X (%) V (%) Z (%) 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 −0.5684 0.9404 0.0000 0.5544 −0.5025 0.0000 −0.8295 2.0127 0.0000 1.2112 −0.8618 0.0000 −0.8590 3.1186 0.0000 1.9270 −1.0807 0.0000 −0.7326 4.2176 0.0000 2.6695 −1.1902 0.0000 −0.4926 5.2983 0.0000 3.4190 −1.2126 0.0000 −0.1670 6.3552 0.0000 4.1671 −1.1663 0.0000 0.2246 7.3896 0.0000 4.9109 −1.0667 0.0000 0.6695 8.4029 0.0000 5.6450 −0.9179 0.0000 1.1607 9.3938 0.0000 6.3720 −0.7326 0.0000 1.6913 10.3651 0.0000 7.0920 −0.5249 0.0000 2.2569 11.3167 0.0000 7.8106 −0.3074 0.0000 2.8506 12.2500 0.0000 8.5278 −0.0912 0.0000 3.4709 13.1651 0.0000 9.2478 0.1193 0.0000 4.1165 14.0648 0.0000 9.9707 0.3200 0.0000 4.7832 14.9476 0.0000 10.6963 0.5137 0.0000 5.4710 15.8136 0.0000 11.4219 0.7004 0.0000 6.1783 16.6641 0.0000 12.1503 0.8800 0.0000 6.9054 17.4992 0.0000 12.8802 1.0526 0.0000 7.6520 18.3146 0.0000 13.6114 1.2183 0.0000 8.4184 19.1133 0.0000 14.3455 1.3769 0.0000 9.2057 19.8908 0.0000 15.0795 1.5256 0.0000 10.0142 20.6459 0.0000 15.8164 1.6674 0.0000 10.8437 21.3771 0.0000 16.5546 1.7993 0.0000 11.6970 22.0817 0.0000 17.2943 1.9228 0.0000 12.5742 22.7568 0.0000 18.0353 2.0351 0.0000 13.4739 23.4010 0.0000 18.7792 2.1376 0.0000 14.3974 24.0088 0.0000 19.5231 2.2302 0.0000 15.3462 24.5800 0.0000 20.2684 2.3116 0.0000 16.3174 25.1091 0.0000 21.0150 2.3804 0.0000 17.3111 25.5947 0.0000 21.7631 2.4393 0.0000 18.3287 26.0312 0.0000 22.5126 2.4856 0.0000 19.3659 26.4144 0.0000 23.2621 2.5193 0.0000 20.4227 26.7414 0.0000 24.0116 2.5404 0.0000 21.4964 27.0081 0.0000 24.7610 2.5488 0.0000 22.5842 27.2116 0.0000 25.5119 2.5446 0.0000 23.6817 27.3478 0.0000 26.2614 2.5249 0.0000 24.7849 27.4165 0.0000 27.0109 2.4927 0.0000 25.8923 27.4165 0.0000 27.7590 2.4449 0.0000 26.9955 27.3449 0.0000 28.5057 2.3818 0.0000 28.0930 27.2060 0.0000 29.2523 2.3046 0.0000 29.1794 26.9969 0.0000 29.9962 2.2106 0.0000 30.2517 26.7246 0.0000 30.7387 2.1011 0.0000 31.3057 26.3877 0.0000 31.4769 1.9748 0.0000 32.3387 25.9933 0.0000 32.2138 1.8330 0.0000 33.3492 25.5428 0.0000 32.9464 1.6730 0.0000 34.3359 25.0404 0.0000 33.6763 1.4976 0.0000 35.2959 24.4902 0.0000 34.4005 1.3053 0.0000 36.2293 23.8965 0.0000 35.1205 1.0948 0.0000 37.1360 23.2621 0.0000 35.8363 0.8674 0.0000 38.0146 22.5912 0.0000 36.5451 0.6246 0.0000 38.8679 21.8866 0.0000 37.2482 0.3635 0.0000 39.6946 21.1512 0.0000 37.9458 0.0856 0.0000 40.4946 20.3877 0.0000 38.6349 −0.2091 0.0000 41.2708 19.5989 0.0000 39.3184 −0.5193 0.0000 42.0216 18.7862 0.0000 39.9921 −0.8463 0.0000 42.7487 17.9511 0.0000 40.6588 −1.1902 0.0000 43.4518 17.0978 0.0000 41.3171 −1.5495 0.0000 44.1340 16.2262 0.0000 41.9669 −1.9256 0.0000 44.7950 15.3392 0.0000 42.6069 −2.3158 0.0000 45.4350 14.4367 0.0000 43.2371 −2.7214 0.0000 46.0554 13.5216 0.0000 43.8589 −3.1425 0.0000 46.6575 12.5939 0.0000 44.4694 −3.5762 0.0000 47.2428 11.6549 0.0000 45.0715 −4.0253 0.0000 47.8126 10.7047 0.0000 45.6624 −4.4871 0.0000 48.3656 9.7475 0.0000 46.2421 −4.9615 0.0000 48.9045 8.7805 0.0000 46.8133 −5.4499 0.0000 49.4295 7.8064 0.0000 47.3719 −5.9496 0.0000 49.9418 6.8254 0.0000 47.9207 −6.4604 0.0000 50.4414 5.8387 0.0000 48.4596 −6.9826 0.0000 50.9298 4.8464 0.0000 48.9874 −7.5159 0.0000 51.4084 3.8485 0.0000 49.5053 −8.0577 0.0000 51.8772 2.8464 0.0000 50.0119 −8.6106 0.0000 52.3376 1.8400 0.0000 50.5088 −9.1721 0.0000 52.7881 0.8295 0.0000 50.9972 −9.7419 0.0000 53.2302 −0.1839 0.0000 51.4744 −10.3201 0.0000 53.6667 −1.2014 0.0000 51.9418 −10.9068 0.0000 54.0934 −2.2218 0.0000 52.4007 −11.5005 0.0000 54.5144 −3.2450 0.0000 52.8499 −12.0998 0.0000 54.9285 −4.2709 0.0000 53.2920 −12.7075 0.0000 55.3369 −5.2997 0.0000 53.7243 −13.3195 0.0000 55.7383 −6.3299 0.0000 54.1481 −13.9384 0.0000 56.1341 −7.3629 0.0000 54.5650 −14.5616 0.0000 56.5257 −8.3987 0.0000 54.9734 −15.1918 0.0000 56.9131 −9.4345 0.0000 55.3748 −15.8248 0.0000 57.2962 −10.4717 0.0000 55.7664 −16.4648 0.0000 57.6780 −11.5103 0.0000 56.1496 −17.1104 0.0000 58.0584 −12.5504 0.0000 56.5229 −17.7602 0.0000 58.4387 −13.5890 0.0000 56.8850 −18.4157 0.0000 58.8205 −14.6276 0.0000 57.2387 −19.0782 0.0000 59.2036 −15.6648 0.0000 57.5812 −19.7448 0.0000 59.5868 −16.7034 0.0000 57.9152 −20.4157 0.0000 59.9672 −17.7420 0.0000 58.2422 −21.0908 0.0000 60.3419 −18.7834 0.0000 58.5650 −21.7687 0.0000 60.7110 −19.8262 0.0000 58.8836 −22.4466 0.0000 61.0731 −20.8719 0.0000 59.2022 −23.1259 0.0000 61.4296 −21.9189 0.0000 59.5180 −23.8066 0.0000 61.7791 −22.9687 0.0000 59.8324 −24.4888 0.0000 62.1244 −24.0200 0.0000 60.1426 −25.1709 0.0000 62.4640 −25.0726 0.0000 60.4542 −25.8530 0.0000 62.5623 −25.3870 0.0000 60.6030 −26.1730 0.0000 62.6704 −25.7421 0.0000 60.7742 −26.5309 0.0000 62.7728 −26.1000 0.0000 60.9496 −26.8860 0.0000 62.8668 −26.4579 0.0000 61.1335 −27.2369 0.0000 62.9511 −26.8200 0.0000 61.3328 −27.5807 0.0000 63.0016 −27.1878 0.0000 61.5756 −27.8937 0.0000 62.9805 −27.5569 0.0000 61.8886 −28.1379 0.0000 62.8668 −27.9092 0.0000 62.2591 −28.2783 0.0000 62.6409 −28.1997 0.0000 62.6409 −28.1997 0.0000 0.3425 1.3446 10.0001 0.3425 1.3446 10.0001 −0.2723 2.2765 10.0001 0.9235 0.8618 10.0001 −0.5811 3.3558 10.0001 1.6000 0.5249 10.0001 −0.6569 4.4787 10.0001 2.3313 0.3256 10.0001 −0.5698 5.6015 10.0001 3.0850 0.2428 10.0001 −0.3663 6.7089 10.0001 3.8443 0.2484 10.0001 −0.0758 7.7966 10.0001 4.5994 0.3228 10.0001 0.2849 8.8619 10.0001 5.3460 0.4533 10.0001 0.7018 9.9075 10.0001 6.0829 0.6316 10.0001 1.1677 10.9335 10.0001 6.8099 0.8463 10.0001 1.6730 11.9384 10.0001 7.5285 1.0905 10.0001 2.2134 12.9251 10.0001 8.2415 1.3516 10.0001 2.7874 13.8949 10.0001 8.9517 1.6155 10.0001 3.3881 14.8465 10.0001 9.6647 1.8765 10.0001 4.0155 15.7813 10.0001 10.3805 2.1277 10.0001 4.6667 16.6992 10.0001 11.0991 2.3678 10.0001 5.3390 17.6017 10.0001 11.8219 2.5979 10.0001 6.0338 18.4887 10.0001 12.5475 2.8169 10.0001 6.7482 19.3575 10.0001 13.2760 3.0274 10.0001 7.4850 20.2094 10.0001 14.0072 3.2281 10.0001 8.2429 21.0403 10.0001 14.7413 3.4190 10.0001 9.0247 21.8515 10.0001 15.4781 3.6001 10.0001 9.8289 22.6375 10.0001 16.2178 3.7685 10.0001 10.6584 23.3982 10.0001 16.9602 3.9257 10.0001 11.5131 24.1309 10.0001 17.7041 4.0702 10.0001 12.3946 24.8326 10.0001 18.4508 4.2022 10.0001 13.3012 25.4993 10.0001 19.2003 4.3201 10.0001 14.2346 26.1281 10.0001 19.9512 4.4239 10.0001 15.1960 26.7148 10.0001 20.7049 4.5152 10.0001 16.1827 27.2551 10.0001 21.4585 4.5909 10.0001 17.1960 27.7450 10.0001 22.2150 4.6527 10.0001 18.2346 28.1814 10.0001 22.9716 4.6976 10.0001 19.2943 28.5576 10.0001 23.7295 4.7271 10.0001 20.3764 28.8706 10.0001 24.4874 4.7411 10.0001 21.4740 29.1148 10.0001 25.2467 4.7369 10.0001 22.5870 29.2874 10.0001 26.0046 4.7145 10.0001 23.7084 29.3871 10.0001 26.7611 4.6752 10.0001 24.8326 29.4109 10.0001 27.5176 4.6176 10.0001 25.9569 29.3576 10.0001 28.2727 4.5418 10.0001 27.0755 29.2285 10.0001 29.0250 4.4464 10.0001 28.1829 29.0250 10.0001 29.7745 4.3327 10.0001 29.2734 28.7513 10.0001 30.5211 4.1994 10.0001 30.3471 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26.3400 30.5674 20.0003 28.6278 5.8780 20.0003 27.4600 30.3906 20.0003 29.3787 5.7503 20.0003 28.5646 30.1380 20.0003 30.1267 5.6015 20.0003 29.6523 29.8138 20.0003 30.8692 5.4303 20.0003 30.7162 29.4222 20.0003 31.6075 5.2380 20.0003 31.7548 28.9660 20.0003 32.3401 5.0260 20.0003 32.7654 28.4537 20.0003 33.0657 4.7916 20.0003 33.7478 27.8867 20.0003 33.7843 4.5376 20.0003 34.7022 27.2734 20.0003 34.4945 4.2625 20.0003 35.6258 26.6151 20.0003 35.1977 3.9678 20.0003 36.5198 25.9176 20.0003 35.8924 3.6534 20.0003 37.3858 25.1849 20.0003 36.5788 3.3207 20.0003 38.2237 24.4200 20.0003 37.2539 2.9685 20.0003 39.0335 23.6270 20.0003 37.9219 2.5993 20.0003 39.8167 22.8073 20.0003 38.5788 2.2120 20.0003 40.5760 21.9638 20.0003 39.2244 1.8091 20.0003 41.3086 21.0978 20.0003 39.8616 1.3895 20.0003 42.0174 20.2136 20.0003 40.4876 0.9544 20.0003 42.7038 19.3097 20.0003 41.1037 0.5053 20.0003 43.3690 18.3918 20.0003 41.7087 0.0407 20.0003 44.0118 17.4571 20.0003 42.3023 −0.4365 20.0003 44.6364 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9.5370 59.9994 10.8254 27.9976 59.9994 16.5715 9.6647 59.9994 11.6942 28.6825 59.9994 17.3223 9.7700 59.9994 12.5925 29.3309 59.9994 18.0760 9.8528 59.9994 13.5174 29.9373 59.9994 18.8325 9.9131 59.9994 14.4732 30.4973 59.9994 19.5905 9.9496 59.9994 15.4557 31.0053 59.9994 20.3484 9.9622 59.9994 16.4648 31.4573 59.9994 21.1077 9.9510 59.9994 17.5006 31.8475 59.9994 21.8656 9.9173 59.9994 18.5589 32.1731 59.9994 22.6207 9.8584 59.9994 19.6354 32.4271 59.9994 23.3744 9.7756 59.9994 20.7273 32.6054 59.9994 24.1253 9.6675 59.9994 21.8291 32.7050 59.9994 24.8733 9.5356 59.9994 22.9351 32.7247 59.9994 25.6158 9.3812 59.9994 24.0396 32.6615 59.9994 26.3527 9.2029 59.9994 25.1372 32.5169 59.9994 27.0839 9.0008 59.9994 26.2207 32.2952 59.9994 27.8081 8.7777 59.9994 27.2874 31.9976 59.9994 28.5253 8.5306 59.9994 28.3302 31.6299 59.9994 29.2355 8.2640 59.9994 29.3478 31.1962 59.9994 29.9373 7.9763 59.9994 30.3387 30.7036 59.9994 30.6306 7.6689 59.9994 31.3001 30.1548 59.9994 31.3155 7.3419 59.9994 32.2306 29.5569 59.9994 31.9906 6.9966 59.9994 33.1317 28.9141 59.9994 32.6573 6.6345 59.9994 34.0019 28.2306 59.9994 33.3127 6.2541 59.9994 34.8440 27.5120 59.9994 33.9598 5.8569 59.9994 35.6552 26.7597 59.9994 34.5970 5.4457 59.9994 36.4398 25.9793 59.9994 35.2244 5.0190 59.9994 37.1977 25.1723 59.9994 35.8419 4.5783 59.9994 37.9290 24.3428 59.9994 36.4496 4.1250 59.9994 38.6363 23.4909 59.9994 37.0475 3.6576 59.9994 39.3199 22.6207 59.9994 37.6356 3.1790 59.9994 39.9809 21.7336 59.9994 38.2139 2.6878 59.9994 40.6195 20.8298 59.9994 38.7837 2.1867 59.9994 41.2371 19.9119 59.9994 39.3437 1.6758 59.9994 41.8364 18.9813 59.9994 39.8939 1.1537 59.9994 42.4160 18.0381 59.9994 40.4357 0.6232 59.9994 42.9803 17.0866 59.9994 40.9690 0.0828 59.9994 43.5276 16.1237 59.9994 41.4925 −0.4660 59.9994 44.0596 15.1539 59.9994 42.0076 −1.0218 59.9994 44.5775 14.1756 59.9994 42.5157 −1.5860 59.9994 45.0813 13.1904 59.9994 43.0139 −2.1572 59.9994 45.5740 12.1995 59.9994 43.5052 −2.7355 59.9994 46.0540 11.2030 59.9994 43.9880 −3.3207 59.9994 46.5228 10.2008 59.9994 44.4638 −3.9116 59.9994 46.9817 9.1931 59.9994 44.9312 −4.5081 59.9994 47.4294 8.1812 59.9994 45.3915 −5.1102 59.9994 47.8687 7.1650 59.9994 45.8463 −5.7194 59.9994 48.2996 6.1461 59.9994 46.2926 −6.3313 59.9994 48.7207 5.1229 59.9994 46.7319 −6.9503 59.9994 49.1361 4.0969 59.9994 47.1642 −7.5734 59.9994 49.5432 3.0681 59.9994 47.5908 −8.1994 59.9994 49.9432 2.0365 59.9994 48.0119 −8.8310 59.9994 50.3361 1.0021 59.9994 48.4245 −9.4668 59.9994 50.7235 −0.0351 59.9994 48.8330 −10.1068 59.9994 51.1053 −1.0737 59.9994 49.2344 −10.7496 59.9994 51.4814 −2.1151 59.9994 49.6316 −11.3967 59.9994 51.8520 −3.1579 59.9994 50.0218 −12.0465 59.9994 52.2169 −4.2022 59.9994 50.4063 −12.7005 59.9994 52.5776 −5.2478 59.9994 50.7867 −13.3574 59.9994 52.9327 −6.2962 59.9994 51.1614 −14.0170 59.9994 53.2850 −7.3461 59.9994 51.5305 −14.6795 59.9994 53.6316 −8.3959 59.9994 51.8955 −15.3448 59.9994 53.9755 −9.4486 59.9994 52.2548 −16.0115 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4.2878 23.0600 79.9997 9.4177 10.4366 79.9997 4.9488 23.9133 79.9997 10.1573 10.5896 79.9997 5.6422 24.7414 79.9997 10.8914 10.7679 79.9997 6.3664 25.5414 79.9997 11.6240 10.9545 79.9997 7.1187 26.3148 79.9997 12.3567 11.1342 79.9997 7.8977 27.0628 79.9997 13.0949 11.2956 79.9997 8.7019 27.7842 79.9997 13.8360 11.4345 79.9997 9.5286 28.4776 79.9997 14.5827 11.5496 79.9997 10.3805 29.1415 79.9997 15.3321 11.6423 79.9997 11.2563 29.7730 79.9997 16.0844 11.7096 79.9997 12.1574 30.3681 79.9997 16.8381 11.7531 79.9997 13.0823 30.9239 79.9997 17.5932 11.7728 79.9997 14.0339 31.4362 79.9997 18.3483 11.7686 79.9997 15.0079 31.8994 79.9997 19.1034 11.7377 79.9997 16.0072 32.3106 79.9997 19.8557 11.6830 79.9997 17.0262 32.6643 79.9997 20.6066 11.6016 79.9997 18.0662 32.9548 79.9997 21.3547 11.4949 79.9997 19.1231 33.1738 79.9997 22.0986 11.3644 79.9997 20.1926 33.3184 79.9997 22.8368 11.2100 79.9997 21.2705 33.3857 79.9997 23.5709 11.0303 79.9997 22.3498 33.3717 79.9997 24.2979 10.8282 79.9997 23.4249 33.2777 79.9997 25.0193 10.6023 79.9997 24.4902 33.1036 79.9997 25.7323 10.3538 79.9997 25.5400 32.8524 79.9997 26.4369 10.0844 79.9997 26.5702 32.5282 79.9997 27.1344 9.7938 79.9997 27.5765 32.1366 79.9997 27.8235 9.4836 79.9997 28.5548 31.6818 79.9997 28.5029 9.1538 79.9997 29.5050 31.1696 79.9997 29.1723 8.8057 79.9997 30.4271 30.6067 79.9997 29.8334 8.4394 79.9997 31.3169 29.9962 79.9997 30.4846 8.0577 79.9997 32.1787 29.3450 79.9997 31.1261 7.6591 79.9997 33.0096 28.6558 79.9997 31.7576 7.2464 79.9997 33.8124 27.9344 79.9997 32.3808 6.8183 79.9997 34.5886 27.1835 79.9997 32.9927 6.3762 79.9997 35.3366 26.4046 79.9997 33.5963 5.9229 79.9997 36.0595 25.6032 79.9997 34.1900 5.4555 79.9997 36.7584 24.7793 79.9997 34.7738 4.9783 79.9997 37.4321 23.9372 79.9997 35.3493 4.4885 79.9997 38.0847 23.0768 79.9997 35.9163 3.9888 79.9997 38.7163 22.1996 79.9997 36.4735 3.4793 79.9997 39.3269 21.3098 79.9997 37.0223 2.9600 79.9997 39.9192 20.4073 79.9997 37.5626 2.4323 79.9997 40.4932 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18.7904 34.5086 89.9999 21.5863 13.3770 89.9999 19.8557 34.5521 89.9999 22.3301 13.1946 89.9999 20.9196 34.5057 89.9999 23.0684 12.9854 89.9999 21.9779 34.3738 89.9999 23.7982 12.7497 89.9999 23.0207 34.1563 89.9999 24.5196 12.4914 89.9999 24.0452 33.8615 89.9999 25.2326 12.2093 89.9999 25.0460 33.4938 89.9999 25.9358 11.9047 89.9999 26.0200 33.0601 89.9999 26.6305 11.5791 89.9999 26.9660 32.5689 89.9999 27.3141 11.2338 89.9999 27.8825 32.0243 89.9999 27.9892 10.8689 89.9999 28.7695 31.4334 89.9999 28.6530 10.4858 89.9999 29.6285 30.8018 89.9999 29.3071 10.0858 89.9999 30.4594 30.1338 89.9999 29.9499 9.6703 89.9999 31.2636 29.4334 89.9999 30.5843 9.2380 89.9999 32.0412 28.7050 89.9999 31.2075 8.7931 89.9999 32.7948 27.9499 89.9999 31.8208 8.3327 89.9999 33.5247 27.1737 89.9999 32.4257 7.8612 89.9999 34.2335 26.3765 89.9999 33.0194 7.3769 89.9999 34.9198 25.5611 89.9999 33.6033 6.8801 89.9999 35.5879 24.7302 89.9999 34.1787 6.3734 89.9999 36.2349 23.8824 89.9999 34.7458 5.8569 89.9999 36.8651 23.0235 89.9999 35.3030 5.3306 89.9999 37.4770 22.1505 89.9999 35.8517 4.7959 89.9999 38.0735 21.2663 89.9999 36.3921 4.2513 89.9999 38.6546 20.3722 89.9999 36.9240 3.6997 89.9999 39.2216 19.4697 89.9999 37.4489 3.1411 89.9999 39.7746 18.5575 89.9999 37.9640 2.5741 89.9999 40.3135 17.6381 89.9999 38.4735 2.0000 89.9999 40.8413 16.7118 89.9999 38.9746 1.4204 89.9999 41.3564 15.7785 89.9999 39.4686 0.8337 89.9999 41.8616 14.8395 89.9999 39.9556 0.2414 89.9999 42.3571 13.8963 89.9999 40.4357 −0.3565 89.9999 42.8427 12.9461 89.9999 40.9086 −0.9586 89.9999 43.3185 11.9932 89.9999 41.3760 −1.5677 89.9999 43.7873 11.0345 89.9999 41.8364 −2.1797 89.9999 44.2462 10.0731 89.9999 42.2911 −2.7972 89.9999 44.6996 9.1075 89.9999 42.7388 −3.4190 89.9999 45.1431 8.1391 89.9999 43.1810 −4.0450 89.9999 45.5810 7.1664 89.9999 43.6175 −4.6752 89.9999 46.0105 6.1910 89.9999 44.0497 −5.3095 89.9999 46.4329 5.2113 89.9999 44.4750 −5.9468 89.9999 46.8484 4.2302 89.9999 44.8947 −6.5882 89.9999 47.2568 3.2450 89.9999 45.3101 −7.2324 89.9999 47.6596 2.2583 89.9999 45.7213 −7.8794 89.9999 48.0554 1.2688 89.9999 46.1256 −8.5306 89.9999 48.4456 0.2765 89.9999 46.5270 −9.1847 89.9999 48.8316 −0.7172 89.9999 46.9214 −9.8415 89.9999 49.2119 −1.7137 89.9999 47.3129 −10.5012 89.9999 49.5867 −2.7116 89.9999 47.6989 −11.1637 89.9999 49.9558 −3.7123 89.9999 48.0807 −11.8275 89.9999 50.3221 −4.7130 89.9999 48.4582 −12.4956 89.9999 50.6814 −5.7166 89.9999 48.8316 −13.1651 89.9999 51.0379 −6.7215 89.9999 49.1993 −13.8374 89.9999 51.3888 −7.7278 89.9999 49.5642 −14.5111 89.9999 51.7369 −8.7356 89.9999 49.9263 −15.1876 89.9999 52.0793 −9.7461 89.9999 50.2828 −15.8655 89.9999 52.4190 −10.7566 89.9999 50.6365 −16.5462 89.9999 52.7558 −11.7672 89.9999 50.9846 −17.2283 89.9999 53.0913 −12.7791 89.9999 51.3298 −17.9132 89.9999 53.4253 −13.7925 89.9999 51.6681 −18.6010 89.9999 53.7594 −14.8044 89.9999 52.0007 −19.2915 89.9999 54.0934 −15.8164 89.9999 52.3278 −19.9862 89.9999 54.4288 −16.8283 89.9999 52.6449 −20.6838 89.9999 54.7657 −17.8403 89.9999 52.9537 −21.3856 89.9999 55.1025 −18.8522 89.9999 53.2555 −22.0901 89.9999 55.4380 −19.8641 89.9999 53.5502 −22.7975 89.9999 55.7720 −20.8761 89.9999 53.8450 −23.5049 89.9999 56.1018 −21.8894 89.9999 54.1397 −24.2123 89.9999 56.4247 −22.9056 89.9999 54.4359 −24.9196 89.9999 56.7390 −23.9245 89.9999 54.7334 −25.6270 89.9999 57.0464 −24.9463 89.9999 55.0295 −26.3330 89.9999 57.3482 −25.9681 89.9999 55.3257 −27.0404 89.9999 57.6429 −26.9927 89.9999 55.6232 −27.7478 89.9999 57.7327 −27.3099 89.9999 55.7622 −28.0720 89.9999 57.8310 −27.6692 89.9999 55.9236 −28.4369 89.9999 57.9236 −28.0285 89.9999 56.0906 −28.7976 89.9999 58.0092 −28.3906 89.9999 56.2661 −29.1541 89.9999 58.0836 −28.7541 89.9999 56.4555 −29.5050 89.9999 58.1229 −29.1246 89.9999 56.6913 −29.8264 89.9999 58.0963 −29.4952 89.9999 57.0001 −30.0790 89.9999 57.9798 −29.8502 89.9999 57.3734 −30.2236 89.9999 57.7552 −30.1366 89.9999 57.7552 −30.1366 89.9999 1.8920 20.9168 100.0000 1.8920 20.9168 100.0000 1.5074 21.9329 100.0000 2.3130 20.2684 100.0000 1.3684 23.0123 100.0000 2.8435 19.7069 100.0000 1.4330 24.1000 100.0000 3.4611 19.2424 100.0000 1.6562 25.1681 100.0000 4.1404 18.8732 100.0000 2.0000 26.2025 100.0000 4.8590 18.5883 100.0000 2.4379 27.2018 100.0000 5.6029 18.3792 100.0000 2.9502 28.1646 100.0000 6.3622 18.2346 100.0000 3.5229 29.0923 100.0000 7.1271 18.1196 100.0000 4.1488 29.9864 100.0000 7.8934 18.0115 100.0000 4.8169 30.8496 100.0000 8.6598 17.9104 100.0000 5.5243 31.6790 100.0000 9.4275 17.8136 100.0000 6.2752 32.4706 100.0000 10.1938 17.7181 100.0000 7.0780 33.2075 100.0000 10.9616 17.6213 100.0000 7.9327 33.8854 100.0000 11.7279 17.5174 100.0000 8.8338 34.5001 100.0000 12.4928 17.4038 100.0000 9.7728 35.0545 100.0000 13.2549 17.2732 100.0000 10.7454 35.5486 100.0000 14.0156 17.1287 100.0000 11.7461 35.9823 100.0000 14.7721 16.9673 100.0000 12.7693 36.3584 100.0000 15.5244 16.7918 100.0000 13.8121 36.6798 100.0000 16.2739 16.6009 100.0000 14.8690 36.9479 100.0000 17.0192 16.3960 100.0000 15.9371 37.1668 100.0000 17.7602 16.1729 100.0000 17.0150 37.3395 100.0000 18.4971 15.9357 100.0000 18.0971 37.4700 100.0000 19.2269 15.6830 100.0000 19.1848 37.5612 100.0000 19.9526 15.4150 100.0000 20.2740 37.6146 100.0000 20.6726 15.1328 100.0000 21.3645 37.6244 100.0000 21.3856 14.8339 100.0000 22.4536 37.5753 100.0000 22.0929 14.5195 100.0000 23.5386 37.4574 100.0000 22.7933 14.1911 100.0000 24.6137 37.2735 100.0000 23.4866 13.8486 100.0000 25.6747 37.0223 100.0000 24.1716 13.4907 100.0000 26.7176 36.7037 100.0000 24.8495 13.1174 100.0000 27.7379 36.3191 100.0000 25.5190 12.7314 100.0000 28.7330 35.8728 100.0000 26.1814 12.3314 100.0000 29.7001 35.3689 100.0000 26.8355 11.9188 100.0000 30.6376 34.8103 100.0000 27.4811 11.4921 100.0000 31.5415 34.2012 100.0000 28.1183 11.0542 100.0000 32.4145 33.5457 100.0000 28.7457 10.6037 100.0000 33.2552 32.8510 100.0000 29.3660 10.1405 100.0000 34.0622 32.1169 100.0000 29.9766 9.6661 100.0000 34.8384 31.3506 100.0000 30.5787 9.1805 100.0000 35.5837 30.5548 100.0000 31.1724 8.6850 100.0000 36.2995 29.7309 100.0000 31.7576 8.1798 100.0000 36.9858 28.8846 100.0000 32.3345 7.6647 100.0000 37.6468 28.0158 100.0000 32.9029 7.1397 100.0000 38.2812 27.1288 100.0000 33.4629 6.6064 100.0000 38.8932 26.2263 100.0000 34.0145 6.0632 100.0000 39.4813 25.3070 100.0000 34.5577 5.5131 100.0000 40.0483 24.3765 100.0000 35.0938 4.9559 100.0000 40.5957 23.4319 100.0000 35.6201 4.3902 100.0000 41.1248 22.4775 100.0000 36.1409 3.8176 100.0000 41.6357 21.5147 100.0000 36.6531 3.2393 100.0000 42.1283 20.5420 100.0000 37.1584 2.6527 100.0000 42.6069 19.5610 100.0000 37.6567 2.0618 100.0000 43.0687 18.5729 100.0000 38.1479 1.4639 100.0000 43.5178 17.5792 100.0000 38.6321 0.8618 100.0000 43.9529 16.5785 100.0000 39.1107 0.2526 100.0000 44.3740 15.5721 100.0000 39.5809 −0.3607 100.0000 44.7824 14.5602 100.0000 40.0469 −0.9783 100.0000 45.1782 13.5440 100.0000 40.5058 −1.6000 100.0000 45.5627 12.5237 100.0000 40.9592 −2.2274 100.0000 45.9375 11.4991 100.0000 41.4055 −2.8576 100.0000 46.3010 10.4717 100.0000 41.8476 −3.4934 100.0000 46.6575 9.4401 100.0000 42.2841 −4.1306 100.0000 47.0070 8.4071 100.0000 42.7150 −4.7734 100.0000 47.3508 7.3713 100.0000 43.1417 −5.4190 100.0000 47.6877 6.3341 100.0000 43.5627 −6.0675 100.0000 48.0217 5.2955 100.0000 43.9796 −6.7187 100.0000 48.3516 4.2555 100.0000 44.3922 −7.3727 100.0000 48.6772 3.2155 100.0000 44.8006 −8.0296 100.0000 49.0014 2.1741 100.0000 45.2034 −8.6892 100.0000 49.3200 1.1312 100.0000 45.6034 −9.3517 100.0000 49.6372 0.0870 100.0000 45.9992 −10.0156 100.0000 49.9516 −0.9572 100.0000 46.3908 −10.6823 100.0000 50.2618 −2.0028 100.0000 46.7796 −11.3517 100.0000 50.5691 −3.0499 100.0000 47.1642 −12.0226 100.0000 50.8737 −4.0969 100.0000 47.5459 −12.6949 100.0000 51.1741 −5.1453 100.0000 47.9249 −13.3700 100.0000 51.4730 −6.1952 100.0000 48.2996 −14.0465 100.0000 51.7663 −7.2450 100.0000 48.6716 −14.7244 100.0000 52.0583 −8.2963 100.0000 49.0393 −15.4037 100.0000 52.3474 −9.3489 100.0000 49.4056 −16.0844 100.0000 52.6351 −10.4001 100.0000 49.7705 −16.7680 100.0000 52.9214 −11.4528 100.0000 50.1326 −17.4515 100.0000 53.2064 −12.5054 100.0000 50.4905 −18.1364 100.0000 53.4927 −13.5581 100.0000 50.8442 −18.8241 100.0000 53.7804 −14.6107 100.0000 51.1923 −19.5147 100.0000 54.0695 −15.6620 100.0000 51.5334 −20.2080 100.0000 54.3615 −16.7132 100.0000 51.8688 −20.9056 100.0000 54.6548 −17.7645 100.0000 52.1972 −21.6059 100.0000 54.9495 −18.8143 100.0000 52.5200 −22.3077 100.0000 55.2415 −19.8655 100.0000 52.8400 −23.0123 100.0000 55.5306 −20.9168 100.0000 53.1600 −23.7168 100.0000 55.8155 −21.9694 100.0000 53.4786 −24.4214 100.0000 56.0962 −23.0235 100.0000 53.8001 −25.1246 100.0000 56.3741 −24.0789 100.0000 54.1243 −25.8277 100.0000 56.6464 −25.1344 100.0000 54.4513 −26.5281 100.0000 56.9061 −26.1941 100.0000 54.7825 −27.2270 100.0000 57.1601 −27.2551 100.0000 55.1152 −27.9246 100.0000 57.2359 −27.5681 100.0000 55.2625 −28.2334 100.0000 57.3201 −27.9204 100.0000 55.4281 −28.5815 100.0000 57.4015 −28.2741 100.0000 55.5966 −28.9267 100.0000 57.4773 −28.6278 100.0000 55.7692 −29.2706 100.0000 57.5461 −28.9843 100.0000 55.9559 −29.6074 100.0000 57.5840 −29.3436 100.0000 56.1776 −29.9190 100.0000 57.5517 −29.7043 100.0000 56.4654 −30.1731 100.0000 57.4268 −30.0439 100.0000 56.8106 −30.3415 100.0000 57.1882 −30.3106 100.0000 57.1882 −30.3106 100.0000

TABLE II SUCTION-SIDE PRESSURE-SIDE X(%) Y(%) Z(%) X(%) Y(%) Z(%) 62.4640 −25.0726 0.0000 60.4542 −25.8530 0.0000 62.5623 −25.3870 0.0000 60.6030 −26.1730 0.0000 62.6704 −25.7421 0.0000 60.7742 −26.5309 0.0000 62.7728 −26.1000 0.0000 60.9496 −26.8860 0.0000 62.8668 −26.4579 0.0000 61.1335 −27.2369 0.0000 62.9511 −26.8200 0.0000 61.3328 −27.5807 0.0000 63.0016 −27.1878 0.0000 61.5756 −27.8937 0.0000 62.9805 −27.5569 0.0000 61.8886 −28.1379 0.0000 62.8668 −27.9092 0.0000 62.2591 −28.2783 0.0000 62.6409 −28.1997 0.0000 62.6409 −28.1997 0.0000 61.9433 −25.2804 10.0001 59.9236 −26.0326 10.0001 62.0374 −25.5961 10.0001 60.0682 −26.3541 10.0001 62.1412 −25.9526 10.0001 60.2338 −26.7148 10.0001 62.2381 −26.3105 10.0001 60.4051 −27.0713 10.0001 62.3265 −26.6698 10.0001 60.5833 −27.4249 10.0001 62.4065 −27.0320 10.0001 60.7784 −27.7702 10.0001 62.4514 −27.3997 10.0001 61.0156 −28.0860 10.0001 62.4261 −27.7688 10.0001 61.3258 −28.3344 10.0001 62.3068 −28 1197 10.0001 61.6935 −28.4790 10.0001 62.0767 −28.4046 10.0001 62.0767 −28.4046 10.0001 61.4268 −25.4965 20.0003 59.3987 −26.2221 20.0003 61.5167 −25.8123 20.0003 59.5377 −26.5435 20.0003 61.6149 −26.1674 20.0003 59.6977 −26.9042 20.0003 61.7061 −26.5239 20.0003 59.8633 −27.2607 20.0003 61.7903 −26.8832 20.0003 60.0373 −27.6144 20.0003 61.8647 −27.2439 20.0003 60.2268 −27.9597 20.0003 61.9040 −27.6102 20.0003 60.4612 −28.2755 20.0003 61.8717 −27.9765 20.0003 60.7658 −28.5253 20.0003 61.7468 −28.3232 20.0003 61.1321 −28.6713 20.0003 61.5124 −28.6025 20.0003 61.5124 −28.6025 20.0003 60.9103 −25.7183 30.0004 58.8724 −26.4214 30.0004 60.9959 −26.0312 30.0004 59.0071 −26.7428 30.0004 61.0886 −26.3849 30.0004 59.1629 −27.1007 30.0004 61.1756 −26.7400 30.0004 59.3243 −27.4572 30.0004 61.2542 −27.0979 30.0004 59.4942 −27.8081 30.0004 61.3244 −27.4572 30.0004 59.6808 −28.1534 30.0004 61.3581 −27.8207 30.0004 59.9096 −28.4678 30.0004 61.3202 −28.1829 30.0004 60.2114 −28.7162 30.0004 61.1896 −28.5253 30.0004 60.5735 −28.8636 30.0004 60.9524 −28.7990 30.0004 60.9524 −28.7990 30.0004 60.3812 −25.9344 40.0006 58.3419 −26.6291 40.0006 60.4654 −26.2474 40.0006 58.4738 −26.9492 40.0006 60.5552 −26.6011 40.0006 58.6268 −27.3085 40.0006 60.6409 −26.9562 40.0006 58.7868 −27.6635 40.0006 60.7180 −27.3127 40.0006 58.9538 −28.0158 40.0006 60.7840 −27.6706 40.0006 59.1377 −28.3597 40.0006 60.8163 −28.0341 40.0006 59.3664 −28.6755 40.0006 60.7756 −28.3962 40.0006 59.6668 −28.9239 40.0006 60.6437 −28.7358 40.0006 60.0275 −29.0699 40.0006 60.4051 −29.0067 40.0006 60.4051 −29.0067 40.0006 59.8338 −26.1407 49.9993 57.7973 −26.8453 49.9993 59.9180 −26.4565 49.9993 57.9306 −27.1681 49.9993 60.0107 −26.8116 49.9993 58.0864 −27.5288 49.9993 60.0977 −27.1695 49.9993 58.2478 −27.8867 49.9993 60.1763 −27.5274 49.9993 58.4163 −28.2418 49.9993 60.2465 −27.8881 49.9993 58.6029 −28.5885 49.9993 60.2801 −28.2544 49.9993 58.8317 −28.9071 49.9993 60.2437 −28.6193 49.9993 59.1349 −29.1583 49.9993 60.1145 −28.9632 49.9993 59.4984 −29.3071 49.9993 59.8787 −29.2397 49.9993 59.8787 −29.2397 49.9993 59.2738 −26.3428 59.9994 57.2443 −27.0685 59.9994 59.3622 −26.6614 59.9994 57.3833 −27.3927 59.9994 59.4591 −27.0207 59.9994 57.5433 −27.7576 59.9994 59.5489 −27.3800 59.9994 57.7089 −28.1197 59.9994 59.6331 −27.7435 59.9994 57.8815 −28.4776 59.9994 59.7075 −28.1085 59.9994 58.0710 −28.8285 59.9994 59.7482 −28.4776 59.9994 58.3040 −29.1513 59.9994 59.7173 −28.8467 59.9994 58.6099 −29.4067 59.9994 59.5952 −29.1990 59.9994 58.9791 −29.5569 59.9994 59.3636 −29.4853 59.9994 59.3636 −29.4853 59.9994 58.7180 −26.5590 69.9996 56.6955 −27.3028 69.9996 58.8092 −26.8776 69.9996 56.8387 −27.6271 69.9996 58.9089 −27.2383 69.9996 57.0043 −27.9920 69.9996 59.0029 −27.5990 69.9996 57.1741 −28.3541 69.9996 59.0900 −27.9625 69.9996 57.3496 −28.7120 69.9996 59.1685 −28.3274 69.9996 57.5433 −29.0629 69.9996 59.2121 −28.6979 69.9996 57.7777 −29.3857 69.9996 59.1868 −29.0699 69.9996 58.0864 −29.6397 69.9996 59.0703 −29.4250 69.9996 58.4570 −29.7913 69.9996 58.8429 −29.7155 69.9996 58.8429 −29.7155 69.9996 58.1805 −26.7934 79.9997 56.1608 −27.5428 79.9997 58.2703 −27.1092 79.9997 56.3039 −27.8628 79.9997 58.3699 −27.4656 79.9997 56.4710 −28.2236 79.9997 58.4626 −27.8235 79.9997 56.6408 −28.5801 79.9997 58.5496 −28.1829 79.9997 56.8176 −28.9337 79.9997 58.6268 −28.5450 79.9997 57.0113 −29.2790 79.9997 58.6717 −28.9113 79.9997 57.2457 −29.5962 79.9997 58.6450 −29.2790 79.9997 57.5531 −29.8474 79.9997 58.5285 −29.6299 79.9997 57.9208 −29.9934 79.9997 58.3026 −29.9176 79.9997 58.3026 −29.9176 79.9997 57.6429 −26.9927 89.9999 55.6232 −27.7478 89.9999 57.7327 −27.3099 89.9999 55.7622 −28.0720 89.9999 57.8310 −27.6692 89.9999 55.9236 −28.4369 89.9999 57.9236 −28.0285 89.9999 56.0906 −28.7976 89.9999 58.0092 −28.3906 89.9999 56.2661 −29.1541 89.9999 58.0836 −28.7541 89.9999 56.4555 −29.5050 89.9999 58.1229 −29.1246 89.9999 56.6913 −29.8264 89.9999 58.0963 −29.4952 89.9999 57.0001 −30.0790 89.9999 57.9798 −29.8502 89.9999 57.3734 −30.2236 89.9999 57.7552 −30.1366 89.9999 57.7552 −30.1366 89.9999 57.1601 −27.2551 100.0000 55.1152 −27.9246 100.0000 57.2359 −27.5681 100.0000 55.2625 −28.2334 100.0000 57.3201 −27.9204 100.0000 55.4281 −28.5815 100.0000 57.4015 −28.2741 100.0000 55.5966 −28.9267 100.0000 57.4773 −28.6278 100.0000 55.7692 −29.2706 100.0000 57.5461 −28.9843 100.0000 55.9559 −29.6074 100.0000 57.5840 −29.3436 100.0000 56.1776 −29.9190 100.0000 57.5517 −29.7043 100.0000 56.4654 −30.1731 100.0000 57.4268 −30.0439 100.0000 56.8106 −30.3415 100.0000 57.1882 −30.3106 100.0000 57.1882 −30.3106 100.0000

It will also be appreciated that the airfoil 52 disclosed in the above Table I and the trailing edge profile 61 disclosed in Table II above may be scaled up or down geometrically for use in other similar turbine designs. Consequently, the coordinate values set forth in Table I or Table II may be scaled upwardly or downwardly such that the airfoil profile shape or trailing edge profile remains unchanged. A scaled version of the coordinates in Table I and/or Table II would be represented by X, Y and Z coordinate values of Table I and/or Table II, with the X, Y and Z non-dimensional coordinate values converted to units of distance (e.g., inches), multiplied or divided by a constant number.

An important term in this disclosure is “profile”. The profile is the range of the variation between measured points on an airfoil surface and the ideal position listed in Table I and Table II. The actual profile on a manufactured turbine rotor blade will be different than those in Table I, and the design is robust to this variation meaning that mechanical and aerodynamic function are not impaired. Similarly, the actual profile of the trailing edge on a manufactured turbine rotor blade will be different than those in Table II, and the design is robust to this variation meaning that mechanical and aerodynamic function are not impaired. As noted above, a + or −5% profile tolerance is used herein. The X, Y and Z values are all non-dimensionalized relative to the airfoil height.

The disclosed airfoil shape and the associated trailing edge profile optimizes and is specific to the machine conditions and specifications. It provides a unique profile to achieve 1) interaction between other stages in the turbine 10; 2) aerodynamic efficiency; and 3) normalized aerodynamic and mechanical blade loadings. The disclosed loci of points defined in Table I and/or Table II allow the gas turbine 12 or any other suitable turbine to run in an efficient, safe and smooth manner. As also noted, the disclosed airfoil 52 may be adapted to any scale, as long as 1) interaction between other stages in the pressure turbine 10; 2) aerodynamic efficiency; and 3) normalized aerodynamic and mechanical blade loadings are maintained in the scaled turbine.

The airfoil 52 described herein thus improves overall gas turbine 12 efficiency. Specifically, the airfoil 52 provides a desired turbine efficiency lapse rate (ISO, hot, cold, part load, etc.). The airfoil 52 also meets all aeromechanical and stress requirements. The turbine rotor blade 50 described herein has very specific aerodynamic design requirements, such as an upstream bucket radial back pressure (i.e., work splits) and radial velocity triangles for the downstream turbine rotors 150, 250, 350 and 450. Significant cross-functional design effort was required to meet these design goals. The airfoil 52 of the turbine rotor blade 50 thus is of a specific shape to meet aerodynamic, mechanical, and heat transfer requirements in an efficient and cost-effective manner.

This written description uses examples to disclose the invention, including the best mode, and also to enable any person skilled in the art to practice the invention, including making and using any devices or systems and performing any incorporated methods. The patentable scope of the invention is defined by the claims and may include other examples that occur to those skilled in the art. Such other examples are intended to be within the scope of the claims, if they have structural elements that do not differ from the literal language of the claims or if they include equivalent structural elements with insubstantial differences from the literal languages of the claims. 

What is claimed is:
 1. A turbine rotor blade comprising an airfoil having an airfoil shape, the airfoil shape having a nominal profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in Table I, wherein the Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances expressed in a unit of distance by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in the unit of distance; and wherein X and Y values, when connected by smooth continuing arcs, define airfoil profile sections at each Z value, the airfoil profile sections at Z values being joined smoothly with one another to form a complete airfoil shape.
 2. The turbine rotor blade of claim 1, wherein the turbine rotor blade forms part of a stage of a turbine.
 3. The turbine rotor blade of claim 1, wherein the turbine rotor blade is a first stage turbine rotor blade.
 4. The turbine rotor blade of claim 1, wherein the airfoil shape lies in an envelope within at least one of +/−5% and/or +/−5% of a chord length in a direction normal to any airfoil surface location.
 5. The turbine rotor blade of claim 1, wherein the height of the airfoil is about 1 inch to about 10 inches.
 6. The turbine rotor blade of claim 1, wherein the X, Y and Z values are scalable as a function of the same constant or number to provide a scaled-up or scaled-down airfoil.
 7. A turbine rotor blade comprising an airfoil having an uncoated nominal suction-side profile substantially in accordance with suction-side Cartesian coordinate values of X, Y and Z set forth in Table I, wherein the Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances expressed in a unit of distance by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in the unit of distance; and wherein X and Y values, when connected by smooth continuing arcs, define suction-side profile sections at each Z value, the suction-side profile sections at the Z values being joined smoothly with one another to form a complete airfoil suction-side shape.
 8. The turbine rotor blade of claim 7, wherein the turbine rotor blade forms part of a stage of a turbine.
 9. The turbine rotor blade of claim 7, wherein the turbine rotor blade is a first stage turbine rotor blade.
 10. The turbine rotor blade of claim 7, wherein the suction-side shape lies in an envelope within at least one of +/−5% and/or +/−5% of a chord length in a direction normal to any airfoil surface location.
 11. The turbine rotor blade of claim 7, wherein the height of the airfoil is about 1 inch to about 10 inches.
 12. The turbine rotor blade of claim 7, wherein the X, Y and Z values are scalable as a function of the same constant or number to provide a scaled-up or scaled-down airfoil with the complete airfoil suction-side shape.
 13. A turbine rotor blade comprising an airfoil having an uncoated nominal trailing edge profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in Table II, wherein the Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances expressed in a unit of distance by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in the unit of distance; and wherein X and Y values, when connected by smooth continuing arcs, define trailing edge profile sections at each Z value, the trailing edge profile sections at the Z values being joined smoothly with one another to form a complete trailing edge shape of the airfoil.
 14. The turbine rotor blade of claim 13, wherein the turbine rotor blade forms part of a stage of a turbine.
 15. The turbine rotor blade of claim 13, wherein the turbine rotor blade is a first stage turbine rotor blade.
 16. The turbine rotor blade of claim 13, wherein the trailing edge shape lies in an envelope within at least one of +/−5% and/or +/−5% of a chord length in a direction normal to any airfoil surface location.
 17. The turbine rotor blade of claim 13, wherein the height of the airfoil is about 1 inch to about 10 inches.
 18. The turbine rotor blade of claim 13, wherein the X, Y and Z values are scalable as a function of the same constant or number to provide a scaled-up or scaled-down airfoil with the nominal trailing edge profile.
 19. The turbine rotor blade of claim 13, wherein X represents a distance parallel to a turbine axis of rotation. 